Solve for $x$ and $y$ using elimination. $\begin{align*}7x+2y &= 5 \\ 8x+4y &= 7\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}-14x-4y &= -10\\ 8x+4y &= 7\end{align*}$ Add the top and bottom equations. $-6x = -3$ Divide both sides by $-6$ and reduce as necessary. $x = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $x$ in the top equation. $7( \dfrac{1}{2})+2y = 5$ $\dfrac{7}{2}+2y = 5$ $2y = \dfrac{3}{2}$ $y = \dfrac{3}{4}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = \dfrac{3}{4}$.